Page 327 - Matrix Analysis & Applied Linear Algebra

5.6 Unitary and Orthogonal Matrices 323

The situation is exactly as depicted in Figure 5.6.1. Notice that (I−Q)x =
T
uu x is the orthogonal projection of x onto the line determined by u and

T
T

uu x = |u x|. This provides a nice interpretation of the magnitude of the
standard inner product. Below is a summary.

Geometry of Elementary Projectors
n×1
Forvectors u, x ∈C such that u =1,
∗
• (I − uu )x is the orthogonal projection of x onto the orthogonal
complement u , the space of all vectors orthogonal to u; (5.6.3)
⊥
∗
• uu x is the orthogonal projection of x onto the one-dimensional
space span {u} ; (5.6.4)
∗
• |u x| represents the length of the orthogonal projection of x onto
the one-dimensional space span {u} . (5.6.5)

In passing, note that elementary projectors are never isometries—they can’t
be because they are not unitary matrices in the complex case and not orthogonal
matrices in the real case. Furthermore, isometries are nonsingular but elementary
projectors are singular.
Example 5.6.2
Problem: Determine the orthogonal projection of x onto span {u} , and then

2 2
find the orthogonal projection of x onto u ⊥ for x = 0 and u = −1 .
1 3
Solution: We cannot apply (5.6.3) and (5.6.4) directly because u =1, but
this is not a problem because

! ⊥

u
u u
⊥

=1, span {u} = span , and u = .

u u u
Consequently, the orthogonal projection of x onto span {u} is given by
 
T T 2
u u uu 1
x = x =  −1  ,
T
u u u u 2
3
and the orthogonal projection of x onto u ⊥ is
 
T T 2
uu uu x 1
I − x = x − =  1  .
T
T
u u u u 2
−1